Question: Simplify. Multiply and remove all perfect squares from inside the square roots. Assume $b$ is positive. $\sqrt{24b^3}\cdot\sqrt{40b^2}\cdot\sqrt{b^2}=$
Solution: Let's start by merging the square roots: $\begin{aligned} \sqrt{24b^3}\cdot\sqrt{40b^2}\cdot\sqrt{b^2} &=\sqrt{24b^3\cdot 40b^2\cdot b^2} \\\\ &=\sqrt{960b^7} \end{aligned}$ Now we remove all perfect squares from inside the square root: $\begin{aligned} \sqrt{960b^7} &=\sqrt{8^2\cdot 3\cdot 5\cdot \left(b^3\right)^2\cdot b} \\\\ &=\sqrt{8^2}\cdot\sqrt{15}\cdot\sqrt{ \left(b^3\right)^2}\cdot \sqrt{b} \\\\ &=8\cdot \sqrt{15}\cdot b^3\cdot \sqrt{b} \\\\ &=8b^3\sqrt{15b} \end{aligned}$ In conclusion, $\sqrt{24b^3}\cdot\sqrt{40b^2}\cdot\sqrt{b^2}=8b^3\sqrt{15b}$